MIDTERM EXAM II - ECON 2311 FALL 2012

This exam is closed-book and closed-notes. You may use your calculator. You have 75 minutes to complete this exam. The total number of points on the exam is 100. The total value for each question is given in brackets. Remember that I will not regrade exams written in pencil, and that all problems with the grade must be brought to my attention before a week after I return the exams. Assumptions and critical values are stated in the appendix. No other formulas will be provided.

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I. (18 pts) Consider y = β0 + β1 x1 + . . . + βk xk + u, and let {βj } be the corresponding OLS estimator. Choose between ’True(T)’, or ’False(F)’. (2 pts each)

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1. If βj is consistent, E[βj ] = βj .

2. Under MLR.1 through MLR.5, the OLS estimator

√
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N (βj − βj ) is asymptotically normal.

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3. If M LR.4 is replaced by M LR.4′ , βj is not consistent.

4. Even though MLR.6 is not true, we can use t−statistic when the sample size is large.

5. Changing the scale of the y variable will lead to a corresponding change in R2 .

6. If y is in the logarithmic form, changing the scale of the y variable will lead to a corresponding change in
R2 .

7. If y is the family income measured in dollars, then it is mostly used in logarithmic form.

8. Standardized coefficient or equivalently beta coefficient measures the change in y when x is changed by one-standard deviation.

9. In the following model: log(y) = β0 + β1 log(x) + u, β1 is the elasticity of y with respect to x.

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II. (30 pts) Multiple Choice Questions (3 pts each)
1. Under MLR.1 through MLR.3 and MLR.4’
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a. βj is unbiased.
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b. βj is consistent.
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c. n(βj − βj ) is asymptotically normal.
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d. βj is BLUE.
2. Suppose the true model is y = 1 + 0.5 x1 + 0.3 x2 + u and Cov(x1 , x2 ) = 0. And the model satisfies
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MLR.1 through MLR.4. Instead, you regress the following model: y = β0 + β1 x1 + u, and β1 is the OLS estimator. Then
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a. β1 greater than 0.5 even when n → ∞.
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b. β1 less than 0.5 even when n → ∞.
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c. β1 gets close to 0.5 when n → ∞.
d. none of the above.
3. Under MLR.1 through MLR.5, the followings are true with the exception of
a.

√ ˆ n(βj − βj ) is asymptotically normal.

b. t-statistic is asymptotically standard normal.
c. σ 2 is consistent.
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d. βj is not asymptotically efficient.
4. Suppose that you estimate the housing price equation as follows: price = 10.0 + 0.9sqrft − 0.7 ∗ bdrooms +
0.001 ∗ bdrooms · sqrft where the price is measured in dollars. Then for a house with the size 3, 000sqrf t, an increase in one more room to the house with the size fixed at 3, 000sqrf t would lead to ( house price. (please clarify the unit.)
5. The followings are all false, with the exception of
a. If x is the number of rooms of a house, it is mostly used in log form.
b. If x is the years of experience, it is mostly used in log form.
c. If y is the interest rate, it is mostly used in log form.
d. If y is the firm’s revenue measured in thousand dollars, it is mostly used in log form.

) change in the

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6. The followings are true about adjusted-R2 , with the exception of
a. Adjusted-R2 always increases when you add a regressor into the model.
b. Adjusted-R2 itself does not provide information about Goodness-of-fit.
c. Adjusted-R2 is useful to compare two non-nested model.
d. If the adjusted-R2 is increased after you add a new regressor, it would be desirable to keep the new regressor in the model.
7. Consider the following two models.
(1)
(2)

ˆ log(y) = 0.15 + 0.1x + 0.05x2

R2 = 0.78,

y = 0.1 + 0.05 log(x)
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adjusted − R2 = 0.70

R2 = 0.75 adjusted − R2 = 0.73

Then
a. Model (1) would be better because it has higher R2 .
b. Model (2) would be better because it has higher adjusted-R2 .
c. We cannot compare.
( Questions 8 through 10)Consider the following model: y = β0 + β1 x1 + β2 x2 + β3 x3 + u
You want to test whether β2 + β3 = 1 using LM test.
8. Write down the restricted model to estimate.
9. Next, after you saved the residual u from the above regression. Then you will regress u on the following
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˜ variables.(Write down the variables.)
10. From the above regression, you get R2 = 0.1 with the sample size 100. Calculate the LM statistic.
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III. (52 pts) We are interested in the following model

log(price) = β0 + β1 log(nox) + β2 log(dist) + β3 rooms + u where price :median housing price,

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nox : nitrous oxide, parts per 100 mill, rooms : avg number of rooms per house dist : weighted distance to 5 employ centers.
The following is the estimation result log(price) =9.75 − 0.95log(nox) − 0.10log(dist) + 0.30rooms
(0.30) (0.13)
R2 = 0.52,

SSR = 40.7,

(0.05)

(0.02)

N = 104

where inside the parentheses are the standard errors of the estimators.
(Questions 1 through 4) Suppose you are interested in whether log(dist) is statistically significant.
1. Specify the null and alternative hypotheses. (3 pts)
2. Calculate the test statistic (3 pts)
3. Do you accept or reject the null hypothesis at 5% significant level? (3 pts)
4. What is your concluding argument? (3 pts, for example you may write either ”log(dist) is statistically significant”, or is statistically insignificant at 5% significant level”. )
(Questions 5 through 8) Now you are interested in whether log(dist) has a negative effect on housing price.
5. Specify the null and alternative hypotheses. (3 pts)
6. Calculate the test statistic. (3 pts)
7. Do you accept or reject the null hypothesis at 5% significant level? (3 pts)
8. What is your concluding argument? (3 pts)
(Questions 9 through 12) You will test whether the coefficient for log(nox) is one.
9. Specify the null and alternative hypotheses. (3 pts)
10. Calculate the test statistic. (3 pts)
11. Do you accept or reject the null hypothesis at 5% significant level? (3 pts)

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12. What is your concluding argument? (3 pts)
(Questions 13 through 16) You have another test that whether both log(nox) and log(dist) do not explain the housing price. We have estimate the following regression. log(price) =7.62 + 0.37rooms
(0.12) (0.02)

R2 = 0.40,

SSR = 50.7,

N = 104

13. Specify the null and alternative hypotheses. (4 pts)
14. Calculate the test statistic. (5 pts)
15. Do you accept or reject the null hypothesis at 5% significant level? (4 pts)
16. What is your concluding argument? (3 pts)